The Application Of Second Order Partial Differential Equations And Calculus Of Variations

In this paper, we will consider the existence of nontrivial solution to the following quasilinear elliptic problem where ?pu= div((|Vu|)p-2?u),1< p< N, p*= Np/(N-p) is Sobolev critical exponent, V,K,W:RN?R and g:RN x R? R are continuous functions, and h(x, u)= m1(W*(|u|)m2)(|u|)m1-2u+m2(W*(|u|)m1)(|u|)m2-2u.This paper gives an application of second order partial differential equations and calculus of variations, which investigates the existence of the nontrivial solu-tion to a quasilinear elliptic equations with Sobolev critical exponent by applying calculus of variations.At first, we outline the history of the calculus of variations'development and advances of solving second order partial differential equations by it and give some preliminary results.Subsequently, according to the equation conditions, we give the variational framework and transfer the problem which study the existence of the solution to a quasilinear elliptic equation into the problem which discusses the existence of the functional's critical point. By the first version of the Mountain Pass Theorem, we obtain a (P.S.)c sequence, and we prove it is bounded. Because the domain is RN, the Sobolev embedding is not compact. So we introduce the Concentration-Compactness Principle and give the estimate of the critical value. Therefore, applying the inequalities, Fatou's lemma, the second version of the Mountain Pass Theorem and so on, we prove the nontrivial existence of the equation we study.